Translations:Causality and stability of digital systems/36/en

We now divide the poles into two sets: those with magnitude greater than one, which we designate as $a_{1}^{-1}$, $a_{2}^{-1}$, $\ldots$, $a_{r}^{-1}$ and those with magnitude less than one, which we designate as $a_{r+{1}^{-1}}$, $a_{r+{2}^{-1}}$, $\ldots$ ,$a_{p^{-1}}$. For the first set, we pick the regions $|Z|<|a_{i}^{-1}|$ as the regions of convergence. For the second set, we pick the regions $|Z|>|a_{i}^{-1}|$ as the regions of convergence. Thus, all the regions include the unit circle. It follows that the stable impulse response is the two-sided signal given by